


Fixed Points

by Bitenomnom



Series: Mathematical Proof [15]
Category: Sherlock (TV)
Genre: Dialogue-Only, Domestic, Gen, M/M, Mathematics, Sherlock and John squabble, Short, Slice of Life, the least explicit sex possible
Language: English
Status: Completed
Published: 2012-09-28
Updated: 2012-09-28
Packaged: 2017-11-15 05:00:40
Rating: Not Rated
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 532
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/523409
Author URL: https://archiveofourown.org/users/Bitenomnom/pseuds/Bitenomnom
Summary: <blockquote class="userstuff">
              <p>The Fixed Point Method of estimating roots involves employing an algorithm until the input is roughly equal to the output.<br/>Sherlock and John's arguments work in much the same way.</p>
            </blockquote>





	Fixed Points

**Author's Note:**

> I was hoping to do a shorter one for today because I've been trying to put some time into Moebius Trip, and, lo and behold, the best idea I could come up with was conducive to brevity. So please excuse the shortness of today's drabble.

The fixed point method consists of the use of the algorithm  
  
xn+1 = F(xn)

 

You continue iterating through this until xn+1 and xn are sufficiently close, or you get to some maximum number of iterations. The value for xn is your approximation for the root. If xn converges to the root r, and F is continuous, then r = F(r). If this is the case, then we say f(x) = x – F(x) has a root at x = r.

 

The Contractive Mapping Theorem says that if you have F that maps [a,b] onto itself (that is, if your inputs are in [a,b], then the outputs are also in that range), and F also satisfies a Lipschitz condition, which says

 

|F(x) – F(y)| < L(x-y) for every (x,y) in your inputs

 

with 0 < L < 1 (in other words we say that f is a contractive mapping function), then x = F(x) has a unique solution r in between a and b, and for every possible initial starting between a and b, xn+1 = F(xn) converges to r with the error estimate

 

|xn – r| < [Ln/(1-L)]|x1-x0|

 

In other words, if we keep iterating over and over and doing so makes the values get closer and closer together, then we can expect that we’ll find a unique solution (unique root) no matter where we start our iteration from, and the error will be no larger than a value depending on L (how much closer the values grow to each other from iteration to iteration) and the difference between your starting point and your first iteration. We can keep on doing this until we’re basically plugging a number into the function and getting the same number back.

 

([You can also look at it this way.](http://imageshack.us/a/img100/796/fixedpointmethod.jpg))

 

***  
  
            “John, I require your assistance immediately.”

            “I’m kind of busy, Sherlock.”

            “John, the flat might blow up if you don’t assist me immediately.”

            “What the _hell_ are you doing, Sherlock?”

            “John, if you assisted me, you might get to see what I’m doing.”

            “Well, whatever it is, you stop it this very instant, Sherlock.”

            “John, you can only make me stop it if you come into the kitchen.”

            “ _Stop it_ , Sherlock.”

            “John, help me.”

            “Sherlock…”

            “John.”

            “No _._ ”

            “No?”

            “ _No._ ”

 

 

 

 

            “Get your legs off me, Sherlock, I’m trying to relax and watch telly.”

            “As am I, in fact. I don’t see how my legs prevent either of us from doing so.”

            “I’m eating dinner, and they’re in my lap.”

            “Stop eating dinner, then.”

            “You know what? Fine. Okay. I don’t sodding care.”

            “That’s what I thought.”

            “Happy now?”

            “Your hand is on my—m-midsection.”

            “Problem?”

            “None at all.”

            “Good.”

            “Good.”

 

 

 

           

            “I really don’t fancy giving Mrs. Hudson a private performance.”

            “She can only hear us if she’s listening, and she’s on the phone right now.”

            “I was trying to say that we could always—”

            “Your room is out of the question, John.”

            “And why, exactly, is that?”

            “No reason.”

            “Sherlock, what the bloody hell are you— _ngh_.”

            “Aah—don’t you mind that now—”

            “Sherlock, you know _mmph_ —”

            “ _John._ ”

            “Ggh—”

            “Oh—”

            “ _Oh—!_ ”


End file.
